let n be Nat; :: thesis: for a, b, c being Integer
for i1, i2, i3 being Element of n holds { p where p is n -element XFinSequence of NAT : a * (p . i1) = (b * (p . i2)) -' c } is diophantine Subset of (n -xtuples_of NAT)
let a, b, c be Integer; :: thesis: for i1, i2, i3 being Element of n holds { p where p is n -element XFinSequence of NAT : a * (p . i1) = (b * (p . i2)) -' c } is diophantine Subset of (n -xtuples_of NAT)
defpred S₁[ Nat, Nat, Integer] means a * $1 = (b * $2) -' $3;
A1: for n being Nat
for i1, i2, i3 being Element of n
for d being Integer holds { p where p is b₁ -element XFinSequence of NAT : S₁[p . i1,p . i2,d * (p . i3)] } is diophantine Subset of (n -xtuples_of NAT) by Th19;
deffunc H₁( Nat, Nat, Nat) -> Integer = c;
A2: for n being Nat
for i1, i2, i3, i4 being Element of n
for d being Integer holds { p where p is b₁ -element XFinSequence of NAT : H₁(p . i1,p . i2,p . i3) = d * (p . i4) } is diophantine Subset of (n -xtuples_of NAT) by Th13;
for n being Nat
for i1, i2, i3, i4, i5 being Element of n holds { p where p is b₁ -element XFinSequence of NAT : S₁[p . i1,p . i2,H₁(p . i3,p . i4,p . i5)] } is diophantine Subset of (n -xtuples_of NAT) from HILB10_3:sch 5(A1, A2);
hence for i1, i2, i3 being Element of n holds { p where p is n -element XFinSequence of NAT : a * (p . i1) = (b * (p . i2)) -' c } is diophantine Subset of (n -xtuples_of NAT) ; :: thesis: verum